let s be State of SCMPDS ; :: thesis: for iloc being Element of NAT
for a being Int_position holds s . a = (s +* (Start-At iloc,SCMPDS )) . a
let iloc be Element of NAT ; :: thesis: for a being Int_position holds s . a = (s +* (Start-At iloc,SCMPDS )) . a
let a be Int_position ; :: thesis: s . a = (s +* (Start-At iloc,SCMPDS )) . a
a in the carrier of SCMPDS ;
then a in dom s by PARTFUN1:def 4;
then A1: ( dom (Start-At iloc,SCMPDS ) = {(IC SCMPDS )} & a in (dom s) \/ (dom (Start-At iloc,SCMPDS )) ) by FUNCOP_1:19, XBOOLE_0:def 3;
a <> IC SCMPDS by SCMPDS_2:52;
then not a in {(IC SCMPDS )} by TARSKI:def 1;
hence s . a = (s +* (Start-At iloc,SCMPDS )) . a by A1, FUNCT_4:def 1; :: thesis: verum